2.8 Summary

In this chapter, GACS processes are characterized in the time domain in the strict and wide senses, at second-and higher-orders. Moreover, a heuristic characterization in the frequency domain is provided. The class of such nonstationary processes includes, as a special case, the ACS processes. Moreover, ACS processes filtered by Doppler channels and communications signals with time-varying parameters are further examples. The problem of estimating second-order statistical functions of (continuous-time) GACS processes is addressed. The cyclic cross-correlogram is proposed as an estimator of the cyclic cross-correlation function of jointly GACS processes and its expected value and covariance are determined for finite data-record length (Theorems 2.4.6 and 2.4.7). It is shown that, for GACS processes satisfying some mixing assumptions expressed in terms of summability of cumulants, the cyclic cross-correlogram is a mean-square consistent (Theorems 2.4.11 and 2.4.13) and asymptotically complex Normal (Theorem 2.4.18) estimator of the cyclic cross-correlation function. Specifically, the covariance and conjugate covariance of the cyclic-cross correlogram are shown to approach zero as the reciprocal of the data-record length, when the data-record length approaches infinity. Moreover, the rate of convergence to zero of the bias is shown to depend on the rate of decay to zero of the Fourier transform of the data-tapering window (Theorem 2.4.12). An asymptotic bound for the covariance uniform with respect to cycle frequencies and lag shifts is also provided (Corollary 2.4.14). The mixing assumptions made are mild and are generally satisfied by processes with finite or practically finite memory. Then, well-known consistency and asymptotic-Normality results for ACS processes are shown to be obtained by specializing the results of this chapter.

It is shown that uniformly sampling a continuous-time GACS process leads to a discrete-time ACS process. Moreover, the class of the continuous-time GACS processes does not have a discrete-time counterpart. That is, discrete-time GACS processes do not exist.

The discrete-time jointly ACS processes obtained by uniformly sampling two continuous-time jointly GACS processes have been considered. It is shown that the discrete-time cyclic cross-correlation function of the jointly ACS processes is an aliased version of the continuous-time cyclic cross-correlation function of the jointly GACS processes (Theorem 2.5.3).

The DT-CCC of the jointly ACS sequences obtained by uniformly sampling two continuous-time jointly GACS process is proposed as an estimator of samples of the continuous-time cyclic cross-correlation function. Its mean value and covariance for finite number of data samples and finite sampling period are derived (Theorems 2.6.2 and 2.6.3). Then, asymptotic expected value and covariance as the data-sample number approaches infinity are derived (Theorems 2.6.4 and 2.6.5) and the DT-CCC is proved to be a mean-square consistent estimator of an aliased version of the cyclic cross-correlation function. Moreover, it is shown to be asymptotically complex Normal (Theorem 2.6.10). The H-CCC is defined and is adopted to analyze the asymptotic behavior of the DT-CCC as the number of data samples approaches infinity and the sampling period approaches zero, provided that the overall data-record length approaches infinity. The H-CCC is proved to be a mean-square consistent and asymptotically complex Normal estimator of the continuous-time cyclic cross-correlation function (Theorems 2.6.16, 2.6.18, and 2.6.20). In addition, it is proved that the rate of decay to zero of bias is equal to the rate of decay to zero of the Fourier transform of the data-tapering window (Theorem 2.6.17). Moreover, it is shown that the asymptotic performance of the H-CCC is the same as that of the continuous-time cyclic cross-correlogram. Thus, no loss in performance is obtained by making the cyclic spectral analysis of (jointly) GACS processes in discrete-time rather than continuous-time. The proved asymptotic Normality can be applied to find confidence intervals and establish statistical test for the presence of generalized almost cyclostationarity.

The example of a GACS signal received when a transmitted ACS signal passes through the Doppler channel existing between a transmitter and a receiver with constant relative radial acceleration is treated both analytically and by a simulation experiment. Numerical results are reported to validate the analytically derived rates of convergence to zero of bias and standard deviation of the cyclic cross-correlogram.